Integral means of δ-subharmonic functions and classes of entirely regular growth

Classes of functions of finite -type -subharmonic in m are investigated. A membership criterion that defines when a function belongs to the class expressed in terms of its integral q-means is established. Classes of entirely regular growth of functions that are -subharmonic in ^m are introduced and the behavior of these functions at infinity and the distribution of Riesz-associated measures are studied.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 3, pp. 408–416, March, 1992.     

Thermorelaxation resonance in the quantum theory of Brownian motion

The Brownian motion of a quantum particle in a thermal reservoir possessing a finite correlation time _c is considered. Non-Markov Langevin equations for a stationary nonequilibrium state are obtained. At low temperaturesT of the thermal reservoir, the correlation time _c =/2T is fairly long. It is shown that allowance for the damping of the particle momentum over the correlation times _c : _c 1, leads to an oscillating temperature dependence of the relaxation coefficient (1/T) in the region of low temperatures of the thermal reservoir.N. I. Lobachevski Nizhegorod State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 92, No. 1, pp. 119–126, July, 1992.     

On Identities in Algebras Q n,λ Generated by Idempotents

We investigate the presence of polynomial identities in the algebras Q _n, generated by n idempotents with the sum e ( and e is the identity of an algebra). We prove that Q _4,2 is an algebra with the standard polynomial identity F ₄, whereas the algebras Q _4,, 2, and Q _n,, n 5, do not have polynomial identities.     

Spreads admitting elliptic regulizations

Throughout this paper, the underlying projective space is 3-dimensional and Pappian. A spreadL admits aregulization , if is a collection of reguli contained inL and if each element ofL, except at most two lines, is contained either in exactly one regulus of or in all reguli of . Replacement of each regulus of by its complementary regulus (exceptional lines remain unchanged) produces thecomplementry congruence L ^c of L with respect to . IfL ^c is an elliptic linear congruence of lines, then we call anelliptic regulization. Applying a method due to Thas and Walker we construct topological spreads of PG(3,) which admit one elliptic and no further regulization. For each of these spreads we determine the group of automorphic collineations. Among others we obtain also spreads which are the complete intersection of a general linear complex of lines and of a cubic complex of lines.In conclusion, I would like to thank H. H{upavlicek} (Vienna) for valuable suggestions in the preparation of this article.     

Increase of stable processes

One says thatt>0 is an increase time for a real-valued path if stays above the level (t) immediately after timet, and below (t) immediately before timet. Dvoretzkyet al.,⁽¹⁰⁾ proved that Brownian motion has no increase times a.s. This result is extended here to (strictly) stable processes. Specifically, the probability that a stable processX possesses increase times is 0 if and only ifP(X ₁0)1/2.     

Banach Disks and Marrelledness Properties of Metrizalbe <Emphasis Type="Italic">c</Emphasis><Subscript>0</Subscript> (Ω, <Emphasis Type="Italic">X</Emphasis>)

Let be a non-empty set and X a metrizable locally convex space. We show that the metrizable locally convex space c₀ (, X) is p-barrelled (totally barrelled) if and only if X is p-barrelled (totally barrelled). Some applications for closed graph theorems are included.     

A bounded law of the iterated logarithm for Hilbert space valued martingales and its application to U-statistics

Summary A bounded law of the iterated logarithm for martingales with values in a separable Hilbert space H is proved. It is then applied to prove invariance principles for U-statistics for independent identically distributed (-valued) random variables {X _j , j1} and a kernel h: ^m H, m2, which is degenerate for the common distribution function of X _j , j1. This extends to general m results of an earlier paper on this subject and even gives new results in the case H=.     

Über die Häufigkeit großer Differenzen konsekutiver Primzahlen

The following theorem is going to be proved. Letp _m be them-th prime and putd _m :=p _m+1–p _m . LetN(,T), 1/21,T3. denote the number of zeros =+i of the Riemann zeta function which fulfill and ||T. Letc2 andh0 be constants such thatN(,T)T ^c(1–) (logT) ^h holds true uniformly in 1/21. Let >0 be given. Then there is some constantK>0 such that      

Influence of critical fluctuations in the structural phase transition of superconductive coupling

The influence of critical fluctuations in the vicinity of the structural phase transition (SPT) on the isotope effect (IE) is investigated in the framework of the anharmonic model of high-temperature superconductivity. It is shown that if the structural transition and the superconducting transition temperatures coincide. IE is determined by the critical exponent . For <1, the IE factor for the T_c superconducting transition temperature (=–log T_c/log M) is equal to the IE factor for the T₀ structural phase-transition temperature (i.e., =. where =–log T₀/log M) and for >1, =&frac; (if the Coulomb repulsion is neglected). This result has a general character and does not depend on the choice of SPT model. This is illustrated by the example of an exactly solvable SPT model with =2.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 107, No. 1, pp. 162–173, April, 1996.     

Circles and spheres in pseudo-Riemannian geometry

Summary A totally umbilical pseudo-Riemannian submanifold with the parallel mean curvature vector field is said to be an extrinsic sphere. A regular curve in a pseudo-Riemannian manifold is called a circle if it is an extrinsic sphere. LetM be ann-dimensional pseudo-Riemannian submanifold of index (0n) in a pseudo-Riemannian manifold with the metricg and the second fundamental formB. The following theorems are proved. For ₀ = +1 or –1, ₁ = +1, –1 or 0 (2–2 ₀+ ₁2n–2–2) and a positive constantk, every circlec inM withg(c, c) = ₀ andg( _c c, _c c) = ₁ k ² is a circle in iffM is an extrinsic sphere. For ₀ = +1 or –1 (–₀n–), every geodesicc inM withg(c, c) = ₀ is a circle in iffM is constant isotropic and B(x,x,x) = 0 for anyx T(M). In this theorem, assume, moreover, that 1n–1 and the first normal space is definite or zero at every point. Then we can prove thatM is an extrinsic sphere. When = 0 orn, this fact does not hold in general.     

Mouvement moyen et système dynamique gaussien

Summary In the situation of the classical mean motion, we haven planets moving in the plane, planetk+1 being a satellite of planetk. A classcal result then states that planetn has a mean motion,i.e. its mean angular speed between time 0 and timet has a limit whent. We show in this article that any real gaussian dynamical system can be interpreted as the limit of this situation, whenn. From a given nonatomic probability measure on [0,], we construct a transformationT of the complex brownian path (B _u)_0u1 which preserves Wiener measure.T is defined as the limit of a sequenceT _n, whereT _n acts as the motion of 2ⁿ planets. In this way we get a real gaussian dynamical system, whose spectral measure is the symetric probability on [-,] obtained from . The transformationT can be inserted in a flow (T ^t) _t, and the orbitstZ _t=B ₁T ^t still have almost surely a mean motion, which is the mean of .     

Approximate Identities, Almost-Periodic Functions and Toeplitz Operators

A sequence {A } of linear bounded operators is called stable if, for all sufficiently large , the inverses of A exist and their norms are uniformly bounded. We consider the stability problem for sequences of Toeplitz operators {T(k a)}, where a(t) is an almost-periodic function on unit circle and k a is an approximate identity. A stability criterion is established in terms of the invertibility of a family of almost-periodic functions. This family of functions depends on the approximate identity used in a very subtle way, and the stability condition is, in general, stronger than the invertibility condition of the Toeplitz operator T(a).     

Quantum Symmetric Algebras

The plus part U ⁺ of a quantum group U _q (g) has been identified by M. Rosso with a subalgebra G _sym of an algebra G which is a quantized version of R. Ree's shuffle algebra. Rosso has shown that G _sym and G – and hence also Hopf algebras which are analogues of quantum groups – can be defined in a much wider context. In this paper we study one of Rosso's quantizations, which depends on a family of parameters t _ij . G _sym is determined by a family of matrices whose coefficients are polynomials in the t _ij . The determinants of the factorize into a number of irreducible polynomials, and our main Theorem 5.2a gives strong information on these factors. This can be regarded as a first step towards the (still very distant!) goal, the classification of the symmetric algebras G _sym which can be obtained by giving special values to the parameters t _ij .     

An Elementary Approach to Quasi-Isometries of Tree x ℝ n

We prove by elementary means a regularity theorem for quasi-isometries of T x ⁿ (where T denotes an infinite tree), and of many other metric spaces with similar combinatorial properties, e.g. Cayley graphs of Baumslag–Solitar groups. For quasi-isometries of T x ⁿ, it states that the image of {x} x ⁿ (xT) is uniformly close to {y} x ⁿ for some yT, and there is a well-defined surjection . Even stronger, the image of a quasi-isometric embedding of ⁿ⁺¹ in T x ⁿ is close to (a geodesic in T)T)x ⁿ.     

Applications of the degree theorem to absolute continuity on Wiener space

Summary Let (,H, P) be an abstract Wiener space and define a shift on byT()=+F() whereF is anH-valued random variable. We study the absolute continuity of the measuresPºT ^–1and ( _F P)ºT ¹ with respect toP using the techniques of the degree theory of Wiener maps, where _F =det₂(1+F) × Exp{–F–1/2|F|²}.The work of the second author was supported by the fund for promotion of research at the Technion     

The spectra of closed interpolated operators

Let (E ₀,E ₁) be a compatible couple of Banach spaces, and letE : 0Re1 be the complex interpolation spaces ofE ₀,E ₁. LetT be a closed linear operator onE ₀+E ₁, then the restrictionT ofT to eachE is closed. If we denote by the extended spectrum ofT inE , then, under appropriate conditions, it is shown that the map is an analytic multifunction in the strip {C0<Re<1}. We use these results to give some applications to the spectral theory of semigroups.     

Some tilings of the plane whose singular points are uncountable and unbounded

Let I be a tiling of the plane such that for every tile T of I there correspond a tile T of I (not necessarily unique) and an integer k(T, T) (depending on T and T), k(T, T)>2, such that T meets T in k(T, T) connected components. Tiles T and T satisfying this condition are called associated tiles in I. Various properties concerning I and its singular points are obtained. First, it is not possible that every tile in I have a unique associated tile. In fact, there exist infinite families of tiles {F} {F _n:n1} such that F is the unique associated tile for every F _n. Next, if x is a singular point of I, then every neighborhood of x contains uncountably many singular points of I. Finally, the set of singular points of I is unbounded.     

A.P.O. rules are asymptotically non deficient for estimation with squared error loss

Summary The problem considered is sequential estimation of the mean of a one-parameter exponential family of distributions with squared error loss for estimation error and a cost c>0 for each of an i.i.d. sequence of potential observations X ₁, X ₂,...A Bayesian approach is adopted, and natural conjugate prior distributions are assumed. For this problem, the asymptotically pointwise optimal (A.P.O.) procedure continues sampling until the posterior variance of is less than c(r₀+n), where n is the sample size and r ₀ is the fictitous sample size implicit in the conjugate prior distribution. It is known that the A.P.O. procedure is Bayes risk efficient, under mild integrability conditions. In fact, the Bayes risk of both the optimal and A.P.O. procedures are asymptotic to 2V ₀ c, as c0, where V ₀ is the prior expectation of the standard deviation of X ₁ given . Here the A.P.O. rule is shown to be asymptotically non-deficient, under stronger regularity conditions: that is, the difference between the Bayes risk of the A.P.O. rule and the Bayes risk of the optimal procedure is of smaller order of magnitude than c, the cost of a single observation, as c0. The result is illustrated in the exponential and Bernoulli cases, and extended to the case of a normal distribution with both the mean and variance unknown.Research supported by the National Science Foundation Grant NSF-MCS-78-02468     

On the homogeneous Riemannian structures of type ℐ1⊕ℐ3

We study homogeneous Riemannian structures belonging to the class ₁₃ of the classification given by Tricerri and Vanhecke. The main result is the following: a connected, simply connected Riemannian manifoldM admits a homogeneous structureT of type ₁₃,T₃, if and only ifM is isometric to a hyperbolic space ⁿ .This work was partially supported by M.P.I.     

A Simple Proof of the Borel Extension Theorem and Weak Compactness of Operators

Let T be a locally compact Hausdorff space and let C ₀(T) be the Banach space of all complex valued continuous functions vanishing at infinity in T, provided with the supremum norm. Let X be a quasicomplete locally convex Hausdorff space. A simple proof of the theorem on regular Borel extension of X-valued -additive Baire measures on T is given, which is more natural and direct than the existing ones. Using this result the integral representation and weak compactness of a continuous linear map u: C ₀(T) X when c ₀ X are obtained. The proof of the latter result is independent of the use of powerful results such as Theorem 6 of [6] or Theorem 3 (vii) of [13].